Warm Up

Solve each equation.

1.

2.

3. 4t – 7 = 8t + 3

4.

5. 2(y – 5) – 20 = 0

x = 7r = 12.2 or -

5.2

n = 17

y = 15

t =−

5

2=−2

1

2

2x −5 =x −4

n2 −7n+12

n2 +n−20=7

11

r −3.5 =8.7

Proof!

Vocabulary

Today we will review properties of equality from algebra and use them to write “algebraic” proofs.

A proof is an argument that uses logic, definitions, properties, and previously proven statements to show that a conclusion is true.An important part of writing a proof is giving justifications to show that every step is valid.

Page 104

The Distributive Property states that a(b + c) = ab + ac.

Remember!

Solve the equation 4m – 8 = 12. Write the justification for each step.

Given equation 4m−8 =12 +8 +8 Addition Property of Equality

4m=20 Simplify

Simplify

Division Property of Equality ÷4 ÷4 m=5

Given equation

Multiplication Property of Equality

Simplify

SimplifySubtraction Property of Equality

Solve the equation . Write the justification for each step.

3

4x +

2

5=−3

1

2

3

4x +

2

5=−3

1

2

20

3

4x +

2

5

⎛

⎝⎜

⎞

⎠⎟ =20 −3

1

2

⎛

⎝⎜

⎞

⎠⎟

15x + 8 =−70

−8 −8

15x =−78

÷15 ÷15

x =−

78

15=−

26

5=−5

1

5Simplify

Division Property of Equality

Like algebra, geometry also uses numbers, variables, and operations. For example, segment lengths and angle measures are numbers. So you can use these same properties of equality to write algebraic proofs in geometry.

A B

AB represents the length , so you can think of AB as a variable representing a number.

Helpful Hint

AB

Write a justification for each step.

NO = NM + MO

4x – 4 = 2x + (3x – 9) Substitution Property of Equality

Segment Addition Post.

4x – 4 = 5x – 9

–4 = x – 9

5 = x

Addition Property of Equality

Subtraction Property of Equality

Simplify.

Write a justification for each step.

m∠ABC =m∠ABD +m∠DBC ∠ + Postulate

8x° = (3x + 5)° + (6x – 16)°

Subst. Prop. of =

x = 11

8x = 9x – 11

–x = –11

Simplify.

Subtr. Prop. of Equality.

Mult. Prop. of Equality.

Key Idea

You learned in Chapter 1 that segments with equal lengths are congruent and that angles with equal measures are congruent. So the Reflexive, Symmetric, and Transitive Properties of Equality each have corresponding properties of congruence.

Page 106

Numbers are equal (=) and f igures are congruent ≅( ).

Remember!

∠QRS ≅∠QRS

m∠1=m∠2 so m∠2 =m∠1

AB ≅CD and CD ≅EF , so AB ≅EF

320 =32 0

Identify the property that justifies each statement.

Ref lexive Prop of ≅

Symmetric Prop of =

Transitive Prop of ≅

Ref lexive Prop of =

Lesson Quiz

Solve the equation and write justification for each step.

6r – 3 = -2(r + 1)

Given

6r – 3 = -2r - 2 Distributive Property

8r – 3 = -2 Addition Property of Equality

8r = 1 Addition Property of Equality

r =

1

8Division Property of Equality

Assignment today is page 108: 16-19, 23-28, 34 and 40-48 .

Remember that homework help is always available at http://www.thinkcentral.com/index.htm

Today’s keyword is “MG7 2-4”.

You need your theorem notebook tomorrow!

Bonus question on-line tonight!!